Restricted convolution inequalities, multilinear operators and applications
Dan-Andrei Geba, Allan Greenleaf, Alex Iosevich, Eyvindur Palsson, Eric Sawyer
Abstract
For
1≤k<n
, we prove that for functions
F,G
on
R
n
, any
k
-dimensional affine subspace
H⊂
R
n
, and
p,q,r≥2
with
1
p
+
1
q
+
1
r
=1
, one has the estimate
||(F∗G)
|
H
||
L
r
(H)
≤
||F||
Λ
H
2,p
(
R
n
)
⋅
||G||
Λ
H
2,q
(
R
n
)
,
where the mixed norms on the right are defined by
||F||
Λ
H
2,p
(
R
n
)
=
(
∫
H
∗
(∫
|
F
^
|
2
d
H
⊥
ξ
)
p
2
dξ)
1
p
,
with
d
H
⊥
ξ
the
(n−k)
-dimensional Lebesgue measure on the affine subspace
H
⊥
ξ
:=ξ+
H
⊥
. Dually, one obtains restriction theorems for the Fourier transform for affine subspaces. Applied to
F(
x
1
,...,
x
m
)=
∏
m
j=1
f
j
(
x
j
)
on $\R^{md}$, the diagonal
H
0
=(x,...,x):x∈
R
d
and suitable kernels
G
, this implies new results for multilinear convolution operators, including
L
p
-improving bounds for measures, an
m
-linear variant of Stein's spherical maximal theorem, estimates for
m
-linear oscillatory integral operators, certain Sobolev trace inequalities, and bilinear estimates for solutions to the wave equation.